Solve alpha=frac{P_{B}-P_{r}}{P_{r}}

$$\alpha = \frac { P _ { B } - P _ { r } } { P _ { r } }$$

Solve for P_B

$$P_{B}=P_{r}\left(\alpha +1\right)$$

$$P_{r}\neq 0$$

Solution Steps

Multiply both sides of the equation by P_{r}.

$$\alpha P_{r}=P_{B}-P_{r}$$

Swap sides so that all variable terms are on the left hand side.

$$P_{B}-P_{r}=\alpha P_{r}$$

Add P_{r} to both sides.

$$P_{B}=\alpha P_{r}+P_{r}$$

Solve for P_r

$$\left\{\begin{matrix}P_{r}=\frac{P_{B}}{\alpha +1}\text{, }&P_{B}\neq 0\text{ and }\alpha \neq -1\\P_{r}\neq 0\text{, }&P_{B}=0\text{ and }\alpha =-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable P_{r} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by P_{r}.

$$\alpha P_{r}=P_{B}-P_{r}$$

Add P_{r} to both sides.

$$\alpha P_{r}+P_{r}=P_{B}$$

Combine all terms containing P_{r}.

$$\left(\alpha +1\right)P_{r}=P_{B}$$

Divide both sides by \alpha +1.

$$\frac{\left(\alpha +1\right)P_{r}}{\alpha +1}=\frac{P_{B}}{\alpha +1}$$

Dividing by \alpha +1 undoes the multiplication by \alpha +1.

$$P_{r}=\frac{P_{B}}{\alpha +1}$$

Variable P_{r} cannot be equal to 0.

$$P_{r}=\frac{P_{B}}{\alpha +1}\text{, }P_{r}\neq 0$$